where \(\bar{v}^{*}=\bar{v}-\frac{\partial }{\partial z}\left(\frac{1}{N^{2}}\overline{v^{\prime }b^{\prime }}\right)\) 
and \(\bar{w}^{*}=\bar{w}+\frac{\partial }{\partial y}\left(\frac{1}{N^{2}}\overline{v^{\prime }b^{\prime }}\right)\) 
are defined as the “residual” mean meridional and vertical velocities,
respectively, which approximate mass-transport by both Eulerian mean
velocities and eddy fluxes, \(N^{2}={\partial \bar{b_{0}}}{\partial z}\) 
is the Brunt-Vaisala frequency, \(\bar{S}\)  is the diabatic heating
term, and \({\bf \mathcal {{F}}}=-\left(u^{\prime }v^{\prime }\right){{\rm \hat{j}}}+\left(\frac{f}{N^{2}}\overline{v^{\prime }b^{\prime }}\right){\rm {\hat{k}}}\,\,\) 
is the Elissan-Plam (EP) flux. \(b_{0}\)  and \(b\)  represent
the mean (zonally- and meridionally-averaged) and the deviation from
the mean of the buoyancy force and \({{\rm \hat{j}}}\)  (\({{\rm \hat{k}}}\) )
is a unit vectors in the meridional (vertical) direction. On Earth,
the diabatic heating term is important [1]}, and therefore
the residual meridional velocities accurately represent the total
meridional transport of mass in Earth's midlatitudes.
